Title:A Semantic Account of Metric Preservation

Abstract: Program sensitivity measures how robust a program is to small changes in its
input, and is a fundamental notion in domains ranging from differential privacy
to cyber-physical systems. A natural way to formalize program sensitivity is in
terms of metrics on the input and output spaces, requiring that an
$r$-sensitive function map inputs that are at distance $d$ to outputs that are
at distance at most $r \cdot d$. Program sensitivity is thus an analogue of
Lipschitz continuity for programs.
Reed and Pierce introduced Fuzz, a functional language with a linear type
system that can express program sensitivity. They show soundness operationally,
in the form of a metric preservation property. Inspired by their work, we study
program sensitivity and metric preservation from a denotational point of view.
In particular, we introduce metric CPOs, a novel semantic structure for
reasoning about computation on metric spaces, by endowing CPOs with a
compatible notion of distance. This structure is useful for reasoning about
metric properties of programs, and specifically about program sensitivity. We
demonstrate metric CPOs by giving a model for the deterministic fragment of
Fuzz.